Sequences and Series
Sequences and Series
This section covers arithmetic and geometric progressions, recurrence relations and sigma notation, with convergence of geometric series as the conceptual highlight. Exam questions tend to combine a reasoning part - showing a sequence is arithmetic, for instance - with a computational part.
Core rules
- AP: u_n = a + (n − 1)d and S_n = (n/2)[2a + (n − 1)d] = (n/2)(a + l).
- GP: u_n = ar^(n−1) and S_n = a(1 − r^n)/(1 − r) for r ≠ 1, while S_∞ = a/(1 − r) only when |r| < 1.
- A recurrence such as u_(n+1) = 3u_n + 2 defines a sequence term by term; to find a closed form, look for a fixed point or test whether the sequence becomes geometric after a shift.
- Sigma notation: the sum from r = 1 to n of (Ar + B) splits into A·Σr + Bn, using Σr = n(n + 1)/2.
Worked example
The first term of an AP is 5 and the 12th term is 38. Find the sum of the first 20 terms. u_12 = a + 11d = 5 + 11d = 38, so 11d = 33 and d = 3. Then S_20 = (20/2)[2(5) + 19(3)] = 10(10 + 57) = 670. Check with the alternative form: the 20th term is 5 + 19(3) = 62, and 20(5 + 62)/2 = 670.
Traps
- Using n instead of n − 1 in the nth-term formula.
- Applying the sum-to-infinity formula when |r| ≥ 1, where the series diverges.
- Sign errors when r is negative; the formula still works, but the terms alternate.
- Treating a recurrence as though it were an explicit formula for the nth term.